Analysis Of Students Mathematical Connection Errors In Trigonometric Identity Problem Solving

PERIÓDICO TCHÊ QUÍMICA
ARTIGO ORIGINAL
Periódico Tchê Química. ISSN 2179-0302. (2020); vol.17 (n°35)
Downloaded from www.periodico.tchequimica.com
825
ANÁLISE DOS ERROS DE CONEXÃO MATEMÁTICA DOS ALUNOS NA RESOLUÇÃO
DE PROBLEMAS DE IDENTIDADE TRIGONOMÉTRICA
ANALYSIS OF STUDENTS' MATHEMATICAL CONNECTION ERRORS IN
TRIGONOMETRIC IDENTITY PROBLEM SOLVING
MARDIYANA1*; USODO, Budi 2; BUDIYONO3; JINGGA, Anisa Astra4; FAHRUDIN, Dwi5
1,2,3 Lecturer at Mathematics Education, Universitas Sebelas Maret
4,5Postgraduate Student at Mathematics Education, Universitas Sebelas Maret
* Correspondence author
e-mail: mardiyana@staff.uns.ac.id
Received 26 May 2020; received in revised form 18 June 2020; accepted 28 June 2020
RESUMO
A identidade trigonométrica é importante no aprendizado de matemática, porque exige que os alunos
pensem crítica, lógica, sistematicamente e completamente. A resolução de problemas de identidade
trigonométrica exige que os alunos relacionem conhecimento conceitual ou conhecimento processual, que depois
é usado em perguntas. Este estudo envolveu alunos da série X do ensino médio, que foram examinados para
descobrir os tipos de erros de conexão matemática e as causas dos erros. Antes das entrevistas baseadas em
tarefas, 36 estudantes foram inicialmente submetidos a um teste. Com base em várias considerações, sete
estudantes (três homens e quatro mulheres) foram selecionados para serem submetidos a uma entrevista
baseada em tarefas. Esta pesquisa empregou um método qualitativo de pesquisa com desenho de estudo de
caso. Os resultados da pesquisa indicam que os erros na conexão com o conhecimento conceitual são mais
comumente o erro de conectar o conceito algébrico. Por outro lado, 86,11% dos estudantes experimentaram
erros na conexão com o conhecimento processual. Esse erro ocorreu quando os alunos trabalharam em
problemas com identidades trigonométricas que raramente encontravam em exercícios. Erros nas conexões
matemáticas da identidade trigonométrica são causados pela falta de compreensão da operação aritmética
algébrica, ênfase no conceito e conhecimento estratégico. Isso mostra que os alunos precisam de uma variedade
de problemas para poderem dominar várias formas de identidades trigonométricas. O resultado desta pesquisa
também reforça o importante papel dos conceitos algébricos como conhecimento prévio no estudo da identidade
trigonométrica.
Palavras-chave: conexão matemática; erro; conhecimento prévio; processual; conceptual
ABSTRACT
The trigonometric identity is essential in learning Mathematics because it requires students to think
critically, logically, systematically, and thoroughly. Solving trigonometric identity problems requires students to
relate conceptual knowledge or procedural knowledge, which then used in questions. This study involved grade
X students of senior high school, which were examined to find out the types of mathematical connections errors
and causes of the errors. Before task-based interviews were conducted, 36 students were first given a test. Based
on several considerations, seven students ( three males and four females) were selected to undergo a task-based
interview. This research employed a qualitative research method with a case study design. The results of the
analysis indicate that the errors in connecting to conceptual knowledge are most commonly the mistake of
connecting the algebraic concept. On the other hand, 86.11% of students experienced errors in connecting to
procedural knowledge. This error happened when the students worked on problems with trigonometric identities,
which they had rarely encountered in exercises. Errors in mathematical connections in trigonometric identity are
caused by the lack of understanding of the algebraic arithmetic operation, emphasis on the concept, and strategic
knowledge. It shows that students need a variety of problems to be able to master various forms of trigonometric
identities. This research's result also reinforces the critical role of algebraic concepts as prior knowledge in
studying trigonometric identity.
Keywords: mathematical connection; error; prior knowledge; procedural; conceptual
DOI: 10.52571/PTQ.v17.n35.2020.70_MARDIYANA_pgs_825_836.pdf

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1. INTRODUCTION:
Trigonometry is one of the earliest
mathematics topics that link algebraic, geometric,
and graphical reasoning; it can serve as an
essential precursor towards understanding pre-
calculus and calculus (Weber, 2005).
Trigonometric identities play a significant role
when students study geometry and calculus at the
next level. The essence of learning trigonometric
identity is not only knowing its knowledge but also
understanding the discovery or derivation of its
properties. The trigonometric identities hone
students’ ability to think deductively, creatively,
and practice the problem-solving skill.
During the process of learning
mathematics, students need the ability to relate
various mathematical concepts as conveyed in
Bruner's connectivity theorem, "in mathematics,
each concept relates to other concepts."
Therefore, students must be allowed to make
mathematical connections to be more successful
in learning mathematics. Mathematical
connections are explained by Hiebert and
Carpenter (1992) as part of structured networks
such as spider webs, "The junctures, or nodes,
can be thought of as pieces of represented
information, and threads between them as the
connections or relationships.” Hiebert and
Carpenter attributed the process of
conceptualizing the conceptual knowledge to the
corresponding structure, which was then
highlighted that knowledge only exists when some
insight can be interconnected. Also, conceptual
understanding can be formed by increasing the
number of connections between these pieces of
knowledge.
Mathematical connections relate with the
connection between one concept to another, one
procedure to another, as well as the relationship
between facts and procedures, we can see that
mathematical connection are related to the ability
to link conceptual knowledge and procedural
knowledge. Students need both applications of
theoretical and procedural knowledge to solve any
problem correctly (Taconis, et al., 2000; Cracolice,
et al., 2008). Thus, we can divide the mathematical
connection errors into two types, namely errors in
connecting to conceptual knowledge and errors in
connecting to procedural knowledge (Table 1).
Procedural knowledge is a knowledge that
focuses on skills and step-by-step procedures
without explicit reference to mathematical ideas
(Marchionda, 2006). Meanwhile, conceptual
knowledge, according to McCormick (1997),
relates to the relationship between various
knowledge, so that when students can identify this
relationship, it can be said that they have a
conceptual understanding. Surf's research (2012)
showed that there is a significant relationship
between theoretical knowledge and procedural
knowledge in solving problems.
The study aimed to find out the types of
students’ mathematical connections errors and
causes of the errors. Insights provided by this
research can serve some useful indications such
as an error in connecting to conceptual or
procedural knowledge, which was done by
students and the one that became our particular
concern. Besides, it is expected to be useful as an
input for teachers about the importance of making
mathematical connections for students so that
they can be considered in teachers’ plans and
learning instruction.
2. LITERATURE REVIEW:
Many studies have concerned about
misconceptions and making errors (Bush, 2011;
Dhlamini & Kibirige, 2014; Sarwadi & Shahrill,
2014; Mohyuddin & Khalil, 2016; Ndemo &
Ndemo, 2018; Sudihartinih & Purniati, 2020). A
few researchers have also mentioned students’
misconceptions and errors in trigonometry, such
as Orhun (2015) and Usman & Hussaini (2017).
Usman & Hussaini’s research is based on
Newman Error Hierarchical Model, which differs
from this research in terms of research method,
which was used. Orhun’s research is about the
difficulties faced by students in using trigonometry
for solving problems in trigonometry. Although
Orhun studied the students’ ability to develop the
existing concept in trigonometry, the focus of his
research is the trigonometric ratio. On the other
hand, Brown (2006) studied students’
understanding of sine and cosine. He found out a
framework, called trigonometric connection.
Although Brown’s research was on trigonometric
link, the focus of the study was only on the rules of
sine and cosine; therefore, this research focuses
on trigonometric identity. The identity of
trigonometry is one of the topics that high school
students should study. Orhun (2015) found that
the students did not develop the concept of
trigonometry with certainty.
Various literature (NCTM, 2000; Mousley,
2004; Blum, et al., 2007) have identified two main
types of mathematical connections. The first is to
recognize and apply mathematics to contexts
outside mathematics (the relationship between
mathematics, other disciplines, or the real world).
The second concerns the interconnections

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between ideas in mathematics.
Moreover, students can understand
mathematics deeply if they have an understanding
of conceptual and procedural knowledge (Wilkins,
2000). Conceptual knowledge focuses on
understanding the relationship between
mathematical ideas and concepts. Meanwhile,
procedural knowledge focuses on symbolism,
skills, rules, and algorithms that used step by step
in completing mathematical tasks. Students must
learn the concepts at once with procedures so that
they can see the relationship. In line with Wilkins,
the NCTM’s Principles and Standards for School
Mathematics (2000, p. 35) states that developing
fluency in mathematical problem solving requires
a balance and connection between conceptual
understanding and computational ability. The
procedures used by the students while solving
mathematical problems show the various levels of
students' conceptual understanding (Ghazali &
Zakaria, 2011). Meanwhile, the issue of
conceptual knowledge is a problem experienced
even by undergraduate students and prospective
teachers (Arjudin, et al., 2016; Dündar & Gündüz ,
2017).
3. MATERIALS AND METHODS:
3.1. Research methods and participants
Qualitative research method with case
study design was used to investigate the grade X
students in a high school located in Kartasura,
Indonesia. The reason for choosing this school
was because the percentage of national
examination results on high school trigonometric
from 2013-2017 was lower than another topic.
Moreover, the results of interviews with teachers
indicated that students have difficulty in studying
the identity of trigonometry than other issues. Also,
the researchers did not teach in the school; it is to
minimize the bias of research, which may occur
due to the subjectivity of the researchers with
students and school staff.
Researchers have received approval to
research the school concerned through a letter of
agreement signed by the Assistant Principal of
Public Relations and Partnership. Moreover, all
participants have agreed to participate in this
study. Researchers kept their identity to be
confidential.
The qualitative approach was appropriate
because the analysis of the learners’ responses to
the given was used to generate theory (Dhlamini
& Kibirige, 2014). The case study method was
chosen in the hope that through the study of a
somewhat unique individual, valuable insights
would be gained (Fraenkel & Wallen, 2000).
3.2. Data collection
This research used task-based interviews
that have been used by researchers in qualitative
research in mathematics education to gain
knowledge about an individual or group of
students’ existing and developing mathematical
knowledge and problem-solving behaviors
(Lerman, 2014). The tasks used in this research
consists of three items with various degrees of
difficulty. It is intended to identify the mathematical
connection errors of upper students and anticipate
the possibility that only a few students who can
answer the questions. Problems given in this
research are as follows:
Problem A (for the first
test):
1) Prove that 𝑠𝑒𝑐 𝑥 −
𝑠𝑒𝑐 𝑥𝑠𝑖𝑛2
𝑥 = 𝑐𝑜𝑠 𝑥 !
2) Known:
• 𝑎 𝑐𝑜𝑠 𝑥 + 𝑡𝑎𝑛 𝑥 =
𝑠𝑒𝑐 𝑥 and
• − 𝑏 − 𝑡𝑎𝑛 𝑥 =
𝑠𝑒𝑐 𝑥,
• determine the
simplest form of
a.b !
3) Prove that
𝑐𝑜𝑠 𝑥
1 + 𝑠𝑖𝑛 𝑥
=
1−𝑠𝑖𝑛 𝑥
𝑐𝑜𝑠 𝑥
!
Problem B (for the
interview):
1) Prove that 𝑐𝑠𝑐 𝑥 −
𝑐𝑠𝑐 𝑥 𝑐𝑜𝑠2
𝑥 = 𝑠𝑖𝑛 𝑥 !
2) Known:
• 𝑎 𝑠𝑖𝑛 𝑥 + 𝑐𝑡𝑔 𝑥 =
𝑐𝑠𝑐 𝑥 and
• − 𝑏 − 𝑐𝑡𝑔 𝑥 = 𝑐𝑠𝑐 𝑥,
• determine the
simplest form of a.b
!
3) Prove that
𝑠𝑖𝑛 𝑥
1 +𝑐𝑜𝑠 𝑥
=
1−𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 𝑥
!
Question number one is a matter of easy
category since it only tests students’ ability to
relate their knowledge of distributive properties (a
matter of pre-condition) and then comparing it to
the concept of trigonometric identity. Problem
number two is a tricky category because it
examines the ability of students to identify the
relationship between the first equation and the
second equation, which can be operated to
produce a trigonometric identity formula. So, this
problem requires students’ ability to relate the
concept of arithmetic algebra and then linking the
results obtained to the idea of trigonometric
identity. Problem number three is a matter of
medium category because it tests the ability of
students in doing algebraic manipulation that can
generate trigonometric identity form associated
with the form of trigonometric identity, which will be
proven.
Before task-based interviews are
conducted, 36 students were given a test first (see
Problem A). Task-based interviews are conducted
for students who have different content analysis
results (different from other students) and fulfill

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almost every indicator, based on these
considerations, seven students ( three males and
four females) were selected to undergo a task-
based interview. Task-based interviews are
carried out to get valid data about errors in
mathematical connections that students do and to
get data about the causes of failures. Accurate
data regarding the purpose of the failure is
obtained by conducting interviews at different
times.
During the task-based interview, the
researchers gave the test to the students in which
the problem of the test was similar to the question
presented in the previous test (see problem B).
The problem is made similar to avoiding the
possibility of students memorize the answers. The
data were in the form of statements about things
done during the fulfillment of tests, errors in
making a mathematical connection, causes of
mistakes, and guidance from the researcher.
3.3. Data analysis
The study used a content analysis
technique that was also used by Luneta (2015)
when researching students' misconceptions. In
this technique, each question is analyzed
according to the content they contain (student
errors); in this case, the students' answers are
indicative of their ability to interact with
trigonometric identity questions. The variables
measured are their responses (misconceptions
and related errors) to the correct answers. The
analysis made inferences to the communication
(student’s answers) by systematically and
objectively identifying specific characteristics of
the student’s errors in the answers. The unit of
analysis was the errors students displayed on
each questions. Errors made by subjects are
grouped and identified its characteristics.
Indicators of the type of mathematical connection
errors can be expressed in Table 1.
4. RESULTS AND DISCUSSION:
The following is the analysis of the
mathematical connection error that students do in
working on the problem of trigonometric identity
along with its causes. A summary of students’
errors on the items is provided in Table 2.
In the transcript, the researcher is referred
to as R meanwhile, the seven students interviewed
are referred to as S.N.2, S.N.5, S.N.6, S.N.12,
S.N.1, S.N.33, and S.N.34 which imply the
students’ serial number in the class. There is also
an abbreviation in the figure such as K.S.17.1. K.S
refers to student errors, 17 refers to the serial
number of students in the class, and 1 refers to the
error number.
4.1. Error in mathematical connection problem
number 1
4.1.1 Type A
In question number 1, four students include
subject number 33, failed in relating knowledge of
algebraic arithmetical operations. During the initial
test, the student performed a reduction operation
on sec 𝑥 – sec 𝑥 sin2
𝑥, which resulted in sin2
𝑥.
When the student was asked to solve a question
for a task-based interview, the student redid the
same mistake. It can be seen from the quotation I
as follows:
Quote I
S.N.33:
1
sin 𝑥
−
1
sin 𝑥
(1 − sin2
𝑥) results in (1-sin2𝑥)
because
1
sin 𝑥
-
1
sin 𝑥
the result is zero.
R: Look,
1
sin 𝑥
multiplied by1- sin2𝑥, right?
S.N.33: Yeah ...
R: Let this a (the researcher designated
1
sin 𝑥
) minus
b (the researcher pointed 1-sin2x) multiplied by a,
what is the result?
S.N.33: a – ab
R: So, is it true if you write it as this (designate
1
sin 𝑥
−
1
sin 𝑥
(1 − sin2
𝑥))? Is it true if the result is 1-sin2x?
S.N.33: No ...
According to quote I, the student thought
that
1
sin 𝑥
-
1
sin𝑥
(1- sin2
x) resulted in 1- sin2
x. It
shows that the student failed in relating the
concept of algebraic form calculation, which had
been learned in junior high school to be applied to
this problem. That algebraic form contains
fractions, Bush (2011) in his research, showed that
students often applied the wrong algorithm when
adding, subtracting, multiplying, or dividing
fractions. When the researcher simplified the
algebraic form, the student was able to answer the
researcher’s question correctly. So, this error
occurred since the student did not understand the
concept of algebraic arithmetic operations.
Problems regarding understanding algebraic
concepts are problems that have attracted the
attention of many researchers (House & Telese,
2008; Nathan & Koellner, 2007; Bush, 2011;
Pournara, et al., 2016), this suggests that the
notion of algebraic concepts are things that are
commonly experienced by students. The results of

829
this research, which focuses on trigonometric
identities, also found students who lacked
understanding of algebraic concepts; this finding
supports the research of Usman & Hussaini (2017)
which focused on the manipulation of
trigonometrical ratios using formula and the right-
angled triangle. They noted that the students’ error
in solving trigonometrical problems was due to
their weaknesses in basic arithmetical operations.
4.1.2 Type B
In question number 1, four students failed
in relating their knowledge about distributive
properties to change an algebraic form. One of
four students who experienced this failure is
subject 33, as seen in the following quotation II:
Quote II
R: In this “𝑎 − 𝑎𝑏” form (𝑎 stands for
1
sin 𝑥
and 𝑏
stands for (1-sin2𝑥)), can you change its shape
based on its distributive properties? (Students
were asked to change “𝑎 − 𝑎𝑏” form into “𝑎(1 −
𝑏)” form)
S.N.33: Yes ..... it becomes 𝑎 − 𝑏.
R: Really? If we apply distributive properties to
this form, it will become “𝑎”. Then, what is inside
the bracket?
S.N.33: Uh, it becomes 1.
R: Okay, 1 and then?
S.N.33: Minus 𝑏.
The student cannot explore her cognitive
ability about distributive properties on
1
sin𝑥
−
1
sin𝑥
(1 − sin2
𝑥) becomes
1
sin𝑥
(1- (1-sin2
x)) or
becomes
1
sin 𝑥
−
1
sin𝑥
+
1
sin𝑥
. sin2
𝑥. It indicates that
the student has failed to relate her prior
knowledge. Gentile & Lalley (2003) revealed that
the mastery of prior knowledge was required by
students in the study of mathematics, it is an
important step in concept development. This
happens because the learning process in
Mathematics is categorized as a widely related
hierarchical learning process (Usman & Hussaini,
2017). If prior knowledge or skills do not exist, the
task of learning becomes more difficult, if the prior
knowledge has been wrongly understood, the
need to learn a new topic results in students for
getting previous misconceptions.
Based on quote I, students still encounter
difficulties and need guidance from the researcher
to apply distributive properties although
1
sin 𝑥
−
1
sin 𝑥
(1 − sin2
𝑥) has been simplified its algebraic
form into a-ab. Thus, we can conclude that the
cause of these errors is the inability to apply their
basic knowledge of distributive properties to new
situations. Mulungye,et al. (2016) stated that the
distributive property says that 𝑎(𝑏 + 𝑐) = 𝑎𝑏 +
𝑎𝑐. Therefore, students use this rule in new
situations that are not appropriate. The distributive
properties of the multiplication toward the sums
are memorized in their minds that they intuitively
misapplied them to a similar situation.
Students also failed to relate their
knowledge about 1- sin2
x = cos2
x, which was then
applied to question number 1. The four students
use similar methods in solving problems. They
changed sec 𝑥 to
1
cos 𝑥
, during the first test, but all
four students made a mistake when changing the
form of csc 𝑥 in the second test. The students did
not know that another form of csc 𝑥 is
1
sin𝑥
although
the researcher had tried to guide them; so, they
still cannot answer the question correctly. The fact
that they can change sec 𝑥 to
1
cos 𝑥
, but cannot
change csc 𝑥 to
1
sin𝑥
Indicates that they memorize
the similarities that exist in trigonometric identities.
Tobias (1993) stated that mathematical anxiety
can cause a person to forget. This is consistent
with the results of further interviews; the subject
number 34 said that he felt depressed and
considered the topic confusing, this anxiety
caused him to forget the trigonometry identities.
Mathematical anxiety is a feeling of tension and
anxiety that interferes the manipulation of
numbers and solving mathematical problems in
various academic situations and everyday life.
Some students tend to experience this anxiety in
exams. Meanwhile, others are only afraid of
mathematical calculations. Wells (1994) identified
anxiety as a significant factor that hinders
students’ reasoning, memory, understanding of
general concepts, and respect to Mathematics.
number 2
4.2.1 Type A
As many as 44.4% of students failed to
relate their knowledge about the concept of
algebraic operation, especially in trigonometric
identities. Students concluded that if 1- sin2
𝑥 =
cos2
𝑥 then 1-sin 𝑥 = cos 𝑥. The student thought
that √1 = √sin2𝑥 + cos2𝑥 produces 1 = sin 𝑥 + cos
𝑥. Students considered that the square root
function was linear. Students tend to over-
regenerate what has been experienced as

830
something ‘true’ (De Bock, Van Dooren, Janssens,
& Verschaffel, 2002)
The failure to relate knowledge of algebraic
arithmetic operations also occurred when
changing the form of 𝑎 cos 𝑥 + tan 𝑥 = sec 𝑥 into 𝑎
= -cos 𝑥 - tan 𝑥 + sec 𝑥. This error occurred since
the student did not understand the concept of
addition, reduction, and multiplication, which are
operated together. The researcher found that
when the algebraic form was still in a complicated
form, the students were not able to cope, but when
the algebraic form was already in a simple form,
the students were able to overcome it.
Researchers found that students can overcome
errors in connecting the concept of algebraic
operations when researchers simplify the
algebraic form. The student was able to answer
the question and followed the direction given by
the researcher when the researcher simplified the
algebraic form and gave direction to him. Thus, the
cause of this error was that the students do not
understand the concept of algebraic operation.
The findings of this research support Norasiah’s
research (2002) which noted that most average
students faced difficulty in performing
trigonometrical operations.
4.2.2 Type B
As many as 52.78% of students did not try
to associate trigonometric identities with other
trigonometric identities. Still, they chose to
elaborate trigonometric identity formulas, which
lead to more complex counting operations. If the
student was able to handle such complexity as
student number 26, he would achieve the simplest
form. If the student was unable to handle it, he
would have difficulty in finding the most
straightforward way; it is experienced by subject
number 17 (Figure 1).
When the students were asked to work on
similar problems and then interviewed about the
results of their work, they reused this strategy. It
can be seen in the following quotation III
Quote III
R: Why did you choose this strategy? Did you
elaborate each of these identities?
S.N.17: If it was not elaborated, the answer
would not be the simplest one ...
R: Oohh… like that … do you know the identity
which has something to do with csc x and ctg x?
Something with a quadratic form?
S.N.17: Yes ...
R: What is it?
S.N.17: Cosecant x is equal to one per sinus x
R: No …something about csc2x…
S.N.17: csc2x =
1
sin2𝑥
R: No…cosecant x is connected with cotangent
x…
S.N.17: I forgot …
R: Do you ever think about this? This is ctg x, and
this is csc x; then, its b is also there ctg x, csc x
… then you look for a itself…b itself which were
then sought the relationship between the two?
S.N.17: No …It is hard to think.
Figure 1. The response of subject number 17
Based on the quotation III, it indicates that
the student failed to associate his knowledge
about sec2
𝑥 – tan2
𝑥 = 1, which was then applied
to the results of multiplication a and b. It happened
because the student thought that a simple form
would be achieved if each trigonometric identity
was altered. The selection of such strategy
resulted in complicated calculations; in this case,
student number 17 was not able to overcome the
complexity, which resulted in a miscalculation (see
K.S.17.3 in Figure 1). Wright (2014) stated that
students need the ability to choose the applicable
formula based on the context of the mathematical
problems they encounter.
As many as 5.6% of students chose the
strategy of eliminating the system of equations,
which caused some more complex counting
operations. Although the students tried to solve
this problem by linking their knowledge of

831
elimination rules, this strategy is not appropriate. It
indicates that the students did not understand the
concept that the two equations are not linear
equations (error type A). Although the students
could change the equations in the form of linear
equations, the number of unknown factors was
more than the sum of the equations. It caused the
system of linear equations not to have a solution.
If it had a solution, it would have an infinite number
of solutions. Besides, the student also failed to
associate the fact that sec2
𝑥 – tan2
𝑥 is equal to 1
(error type B). One of the students using this
strategy was student number 6, as seen in the
following quotation IV:
Quote IV
R: Why did not you try another strategy to solve
this problem? Such as determining the value of a
and b first, then doing the multiplication operation
towards a and b?
S.N.6: Uumm …I did not think like that.
R: Was it because there were two equations,
then you chose to eliminate them?
S.N.6: Yes… if there are two equations, I solve it
by elimination.
Based on quote IV, it shows that the
student eliminated both equations (see K.S.6.1
and K.S.6.2 in Figure 2). The cause was the
student’s lack of understanding of the concept of
the linear equation system. The student assumed
that two equations had to be solved by elimination.
The choice of such a strategy resulted in more
complicated calculations. In this case, subject
number 6 was not able to overcome the complexity
which occurred (see K.S.6.3 and K.S.6.4 in Figure
2) due to complicated forms, resulting in the
student making errors in the calculations.
Figure 2. The response of subject number 6
Based on the results of the analysis on
students number 6 and 17, it shows that the
students did not have good ability to relate one
trigonometric identity with other trigonometric
identities. Based on the interview, all students who
made mistakes in number 3 did not know the
identity value of the sec2
𝑥 – tan2
𝑥, even though,
based on observations, the teacher has given
detailed knowledge about sec2
𝑥 – tan2
𝑥 = 1.
However, the teacher did not provide enough
emphasis on this identity. The given problem was
less varied. It only focused on the relationship of
sin2
𝑥 + cos2
𝑥 = 1. In this case, the researcher
assumed that the cause of the student’s error in
relating his knowledge of sec2
𝑥 – tan2
𝑥 = 1 to be
used in the results of multiplication a and b was the
lack of emphasis on the concept and lacking of
strategic knowledge. This is in accordance with the
results of Lopez's (1996) study which focuses on
mathematical word problems, he also found that
weakness in understanding concepts and lacking
of strategic knowledge result in difficulties in
problem-solving. Moreover, students who were
weak in conceptual understanding were found to
lack arithmetic and procedural skills (Narayanan,
2007).
number 3
4.3.1 Type B
Some students failed to relate their
knowledge of multiplication to convert 1 + cos x
into1-cos2
x. Based on the interview result, it shows
that the student already knew that he had to use
the fact that 1-cos2
x is equal to sin2
x to solve this
problem. However, the student could not exploit
his cognitive ability to change 1+cosx into 1-cos2
x.
It indicates that the student failed in relating his
knowledge of multiplication with a conjugate pair.
The student already knew which identity formula
should be utilized after getting directions from the
researcher. However, the student did not know
how to direct 1 + cos x to be converted into 1-
cos2
x. So, the cause of student’s error was less
skill in doing the algebraic manipulation.
The error which occurred in problem
number 3 is classified as minimal since only seven
students who experienced this problem.
Moreover, no student experienced error type B on
this problem. Based on the observation done by
the researcher, the teacher has given practice and
provided a discussion on this kind of problem.
Besides, some problems with the students’
worksheets also honed their ability to solve this
kind of problem. If the students are accustomed to
solving certain types of problems, they will be able
to solve other problems that are similar in types; it

832
is because the students already understand the
way of settlement. A question or a mathematical
problem is said to be a problem if the person who
faces the challenge feels the gap between where
it is and where it should be but do not know how to
cross the deficit, this results in difficulties in solving
existing problems (Reid & Yang, 2002).
5. CONCLUSION:
The results of this study have provided a
new description of the error of mathematical
connections in solving trigonometric identities.
There are two types of mathematical connection
errors. The first one is errors in connecting to
conceptual knowledge (type A), and the second
one is errors in connecting to procedural
knowledge (type B). Students most commonly do
error type A is the mistake of connecting the
algebraic concept. Although the percentage is
small, we can see that in-depth conceptual
understanding of the linear equation system can
cause connection errors in trigonometric identity
problem solving.
In the second type (type B), an error
occurred most often when the students worked on
problems with trigonometric identities, which they
had rarely encountered in exercises; 86.11% of
students experienced this error. If the given
problem contained common trigonometric
identities encountered in their daily practice, only
30.56% students experience connection errors,
from this fact, we could include that the second
type of error was caused by the lack of emphasis
on the concept and lack of strategic knowledge.
Errors committed by the students in learning
trigonometry can be useful for the teachers in
evaluating their teaching as well as being able to
correct the students’ works appropriately.
6. LIMITATION AND STUDY FORWARD:
In this paper, we just discussed the error of
mathematical connections in solving trigonometric
identities. For the next researcher can discuss the
topics more widely.
7. ACKNOWLEDGEMENTS:
The authors thank to Kartosuro Senior
High School for giving research permission. We
also thank to Universitas Sebelas Maret Surakarta
for budget supporting. The authors are grateful to
the reviewers for helpful comments.
8. REFERENCES:
1. Arjudin, Sutawidjaja, A., Irawan, E. B. &
Sa'dijah, C. (2016). Characterization of
Mathematical Connection Errors in
Derivative Problem Solving. IOSR Journal of
Research & Method in Education, 6, 5, 7-12.
2. Blum, W., Galbraith, P. L., Henn, H.-W. &
Niss, M. (2007). Modelling and applications
in mathematics education. New York:
Springer US.
3. Brown, A. S. (2006). The trigonometric
connection: students‟ understanding of sine
and cosine. Prague, PME30, 228.
4. Bush, S. B. (2011). Analyzing common
algebra-related misconceptions and errors
of middle school students, University of
Louisville: Unpublished Dissertations.
5. Cracolice, M. S., Deming, J. C. & Ehlert, B.
(2008). Concept learning versus problem
solving: a cognitive difference. Journal of
Chemical Education, 85, 6, 873-878.
6. De Bock, D., Van Dooren, W., Janssens, D.
& Verschaffel, L. (2002). Improper use of
linear reasoning: An in-depth study of the
nature and the irresistibility of Secondary
School Students’ Errors. Educational
Studies in Mathematics, 50, 3, 311–334.
7. Dhlamini, Z. B. & Kibirige, I. (2014). Grade 9
Learners’ Errors And Misconceptions In
Addition Of Fractions. Mediterranean
Journal of Social Sciences, 5, 8, 236-244.
8. Dündar, S. & Gündüz , N. (2017).
Justification for the Subject of Congruence
and Similarity in the Context of Daily Life and
Conceptual Knowledge. Journal on
Mathematics Education, 8, 1, 35-54.
9. Fraenkel, J. R. & Wallen, N. E. (2000). How
to Design & Evaluate Research in
Education. Boston: McGraw-Hill
Companies, Inc.
10. Gentile J. R & Lalley, J. P. (2003). Standards
and Mastery Learning Aligning Teaching
and Assessment so All Children can Learn.
Thousand Oaks: Corwin Pres,Inc.
11. Ghazali, N. H. C. & Zakaria, E. (2011).
Students' Procedural and Conceptual
Understanding of Mathematics. Australian
Journal of Basic and Applied Sciences, 5, 7,
684-691.
12. Hiebert, J. & Carpenter, T. P. (1992).
Learning and teaching with understanding.

833
Handbook of Research in Teaching and
Learning of Mathematics. Grouws D. W ed.
New York: Macmilan.
13. House, J. D. & Telese, J. A. (2008).
Relationships between students and
instructional factors and algebra
achievement of students in United States
and Japan: An analysis of TIMSS 2003 data.
Educational Research and Evaluation, 14, 1,
101-112.
14. Lerman, S. (2014). Encyclopedia of
Mathematics Education. London: Springer.
15. Lopez, L. (1996). Helping at-risk students
solve mathematical word problems through
the use of direct instruction and problem
solving strategies, Orlando: Unpublished
Thesis.
16. Luneta, K. (2015). Understanding students’
misconceptions: An analysis of final Grade
12 examination questions in geometry.
Pythagoras, 36, 1, 1-11.
17. Marchionda, H. (2006). Preservice teacher
procedural and conceptual understanding of
fractions and the effects of inquiry based
learning on this understanding, Clemson
University: Unpublished Doctoral
Dissertation.
18. McCormick, R. (1997). Conceptual and
Procedural Knowledge. International
Journal of Technology and Design
Education 7, 141-159.
19. Mohyuddin, R. G., & Khalil, U. (2016).
Misconceptions of Students in Learning
Mathematics at Primary Level. Lahore:
University of Management and Technology.
20. Mousley, J. (2004). An Aspect of
Mathematical Understanding: The Notion of
"Connected Knowing". Bergen, International
Group for the Psychology of Mathematics
Education, 377–384.
21. Mulungye, M., O‘Connor, M. & Ndethiu.
(2016). Sources of student errors and
misconceptions in algebra and effectiveness
of classroom practice remediation in
Machakos County- Kenya. Journal of
Education and Practice, 7, 10, 31-33.
22. Narayanan, L. M. (2007). Analysis of Error in
Addition and Subtraction of Fraction among
Form 2, Universiti Malaya: Unpublished
Thesis.
23. Nathan, M. J. & Koellner, K. (2007). A
Framework for understanding and cultivating
the transition from arithmetic to algebraic
reasoning. Mathematical Thinking and
Learning, 9, 3, 179-192.
24. NCTM. (2000). Principles and Standards for
School Mathematics. Reston, VA: The
National Council of Teachers of
Mathematics, Inc..
25. Ndemo, O., & Ndemo, Z. (2018). Secondary
School Students’ Errors and Misconceptions
in Learning Algebra. Journal of Education
and Learning (EduLearn), 12(4), 690-701.
doi:10.11591/edulearn.v12i4.9956
26. Norasiah, A. (2002). Error type diagnosis in
learning simultaneous equation, s.l.:
Unpublished .
27. Orhun, N. (2015). Students’ Mistakes And
Misconceptions On Teaching Of
Trigonometry. 1 ed. Eskisehir: Anadolu
University.
28. Pournara, C., Hodgen, J., Sanders, Y. &
Adler, J. (2016). Learners’ errors in
secondary algebra: Insights from tracking a
cohort from Grade 9 to Grade 11 on a
diagnostic algebra test. Pythagoras, 37, 1, 1-
10.
29. Reid, N. & Yang, M.-J. (2002). The solving
of problems in chemistry: the more open-
ended problems. Research in Science and
Technological Education, 2, 1, 83-96.
30. Sarwadi, H., & Shahrill, M. (2014).
Understanding Students' Mathematical
Errors and Misconceptions: The Case of
Year 11 Repeating Students. Mathematics
Education Trends and Research, 1-10.
doi:10.5899/2014/metr-00051
31. Sudihartinih, E., & Purniati, T. (2020).
Students' Mistakes and Misconceptions on
the Subject of Conics. International Journal
of Education, 12(2), 92-129.
doi:10.17509/ije.v12i2.19130
32. Surif, J., Ibrahim, N. H. & Mokhtar, M.
(2012). Conceptual and Procedural
Knowledge in Problem Solving. s.l., Elsevier,
Ltd, 416-425.
33. Taconis, R., Ferguson-Hessler &
Broekkamp, H. (2000). Science teaching
problem solving: an overview of
experimental work. Journal of Research in
Science Teaching, 38, 4, 442-468.
34. Tambychik, T. & Meerah, T. S. M. (2010).
Students’ Difficulties in Mathematics
Problem-Solving. Malacca, Elsevier Ltd,

834
142-151.
35. Tobias, S. (1993). Overcoming math anxiety
revised and expanded. New York: Norton.
36. Usman, M. H. & Hussaini, M. M. (2017).
Analysis of students’ error in learning of
trigonometry among Senior Secondary
School students in Zaria Metropolis, Nigeria.
IOSR Journal of Mathematics, 13, 2, 1-4.
37. Weber, K. (2005). Students' understanding
of trigonometric functions. Mathematics
Education Research Journal, 17, 3, 91-112.
38. Wells, D. (1994). Anxiety, insight and
appreciation. Mathematics Teaching,
Volume 147, 8-11.
39. Wilkins, J. (2000). Preparing for the 21 st
century: The status of quantitative literacy in
the United States. School Science and
Mathematics, 100, 8, 405-418.
40. Wright, J. E. (2014). An investigation of
factors affecting student performance in
algebraic word problem solutions, North
Carolina: Gardner-Webb University.

835
Table 1. Indicators for Types of Mathematical Connection Errors
No. Type of Mathematical Connection Errors Indicators
1 Errors in connecting to conceptual knowledge
(Type A)
Applying the concept of calculation incorrectly
Using concepts to prior knowledge inappropriately
2 Errors in connecting to procedural knowledge
(Type B)
Using improper trigonometric rules or formulas
Making a mistake/ unable to do algebraic
manipulation
Making sign errors in count operations
Table 2. Percentage of Students’ Failures on Each Item
Item Description of errors
Percentage of
students’ errors
1 Type
A
Relating knowledge of algebraic arithmetical operations (eg.
1
𝑠𝑖𝑛 𝑥
−
1
𝑠𝑖𝑛 𝑥
(1 − 𝑠𝑖𝑛2
𝑥) produces 1-sin2x)
5.56
Type
B
Relating their knowledge that 1- sin2x is equal with cos2x
13.89
Relating distributive properties to change an algebraic form 8.33
2 Type
A
Relating their knowledge of the algebraic rule (e.g. conclude that √1 =
√𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 produces 1 = 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥)
44.44
Relating their knowledge of the algebraic rule ( a cos x + tan x = sec x
into a = -cos x - tan x+sec)
38.89
Relating their knowledge about the concept of linear equation system 5.56
Type
B
Relating their knowledge that sec2𝑥 – tan2 𝑥 is equal with 1
52.78
Relating distributive properties to change an algebraic form 11.11
3 Type
B
Relating their knowledge of multiplication with conjugate pair to
change an algebraic form
11.11

836
APPENDIX
INTERVIEW GUIDELINES
The purpose of the interview
Obtain information about student mathematical connection errors in solving trigonometric
identity problems and obtain information about the causes of these difficulties.
Interview Conditions
1. The questions adapted to the problem-solving conditions of the students (writing and explanation)
2. If students have difficulty to understand certain questions, the interviewer can provide a guide to
simpler questions without reducing the gist of the problem.
Interview Implementation
1. Provide trigonometric identity test questions.
2. Ask students to read the test questions carefully. Make sure students understand each question
and give students time to work on it.
3. To dig up data regarding errors in connecting to conceptual knowledge (Type A) is done with
a. Asking main questions on the subject, for example, "What is the reason you get this final
result?"
b. Give follow-up questions so that you can explore the information processing done by the
subject in detail when using concepts in initial knowledge, for example, "In this section,
how do you/ what reasons do you get results like this?"
c. Give follow-up questions so that you can explore the information processing carried out by
the subject in detail when applying the concept of calculation, for example, "Please pay
attention to the calculations in this section, is there a miscalculation?"
4. To dig up data regarding errors in connecting to procedural knowledge (Type B) done with
a. Asking main questions on the subject, for example, "How will you solve the problem?"
b. Give follow-up questions so that you can explore the information processing done by the
subject in detail at the stage of determining trigonometric rules or formulas, for example,
"What are the trigonometric identities that you use to solve the problem?"
c. Give follow-up questions so that you can explore the mistakes made by the subject in detail,
for example, " How do you change this algebraic form into another algebraic form?"
d. Give follow-up questions so that you can explore the sign mistake made by the subject in
detail, for example, “Is the sign of the operation/operation of your calculation correct?"
The Periódico Tchê Química (ISSN: 1806-0374; 2179-0302) is an open-access journal since 2004. Journal DOI: 10.52571/PTQ. http://www.tchequimica.com.
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